April 28, 2013

When looking at stock price series, you often hear that such series are random or almost random in nature, and if such was the case, then such series would have little predictability, if any. However, some interesting observations could be made depending on the model used to mimic random stock prices.

The following chart shows a randomly generated price series.

It's easy to build: price varies within a range of +/- $ 5. It uses Excel's rand() function:

Model 1: P(t) = P_{(t-1)} + (RAND() - 0.50)*10

**Model 1: Random Price Move**

(click to enlarge)

Even if the chart spans only 1,000 data points, one would be hard-pressed to say that there is no trend in that chart. Yet, from any one of those points, even if a trend was in formation and could easily be seen, there was no way of predicting what would be the next data point or, for that matter, if the trend could continue, whatever it was in whatever direction. It is sufficient to press F9 and have a totally new and unique price series. Each time you press F9, a new price series will be generated with no memory of any previous iteration.

You can have trends and chart patterns; they are easy to identify when looking at past data. But the real problem is to identify future trends, and there, if you are looking for a profit, you might find that trends are just part of randomness. Based on the above price function, no matter what you would want to use as a predictive function, nothing would help, nothing at all. Should you try trend following, mean reversal, or technical indicators of all sorts, nothing would give you an edge. All you could count on would be luck.

I would add that whatever the trading methodology you would want to use, your expected outcome would be zero profits, and you would also have to make deductions for frictional costs. Whatever portfolio composition rules you might want to use and whatever the correlation between all the stocks under consideration, none of them would have value. Try finding the efficient frontier for a 50 stocks portfolio where all price series were randomly generated as per Model 1. You might find it on past data but be assured that whatever you found will not hold and will have absolutely no value going forward.

The equation of Model 1 says that adopting a symmetrical trading strategy based on point change will tend to produce no profits in the short, mid, and long term as the expected value of this model is no change, whatever the period under consideration.

**A Game Changer**

Now the problem will change with a slight change in the formula:

Model 2: P(t) = P_{(t-1)} * (1 + (RAND() - 0.50)/10)

where prices now change within a range of up to +/- 5%.

**Model 2: Random % Move**

(click to enlarge)

The above could be translated to a portfolio-level consideration, like using a trading strategy taking up to 5% gains and accepting up to 5% stop losses. From the above point, the long-term outcome of such a trading strategy would be total portfolio destruction! Nothing even close to hoping to get even in the long run, as in the first case. It won't be a consideration that the trend seems to degrade with time. It just will.

It is clear that using (RAND() - 0.50) will give a mean of zero, with zero drift and zero correlation. Thereby, there is no predictive value whatsoever, be it a short-, mid-, or long-term scenario; otherwise, the random function would not be random. There is no strategy that can beat heads or tails except pure luck.

The above chart has a negative long-term drift (trend) built in. In fact, long-term, ΣΔP = - P(0), playing the percentage game is more difficult since it is really biased to the downside. The picture is totally different. that you play for points or play for successive percentage moves. In the beginning, both price series might show little difference to the point where it will be hard to tell them apart, but gradually the spread will increase with time.

The first data series says that you might or could reach zero long-term with no certainty (luck playing a major role here) but still have a 50% chance of ending positively. While the second says, you will reach practically zero with an asymptotic probability approaching 1. The difference originates from considering the game as a casino-type game (playing for points) or a compounded return game (playing percent changes). Note that the game itself is a compounded return game, and one can easily design trading strategies that play this type of scenario. It's amazing how often I see this last scenario in naive trading strategies where the author does not seem to understand why his strategy is breaking down going forward.

For those that believe that long term, the game itself is totally random, they are out of luck. Playing the first data series will have an expected profit of zero, no matter how long you want to play. While playing the second series will ultimately have for expected loss, the total capital put into the game.

Both scenarios make the game not worth playing, except if you want to play the lucky side, meaning that all you want to do is gamble and have fun. I would even add, that one should not be surprised to be part of the also-ran after losing their stake, they will have just played for a known outcome which in this case could cost up to the total capital available for play.

**Not Totally Random**

The above should make a good point for the case that prices are not totally random, that they have an underlying long-term drift: P(t) = μdt + σdw where the long-term drift component might be small but still there. For instance, the historical long-term drift has been about $0.04 per day on a $100 stock, while its variance might be in the order of $2.00 per day and often higher.

It all comes down to how are you going to trade and on what basis will your trading strategy evolve over time. I find it easy to design trading strategies with positive expectancy: Σ(**H**.*Δ**P**) > 0, but to be consistent, I also think that price series are not totally random. And it is also why I look at the long term when designing and backtesting my trading strategies. Should I not know how my trading strategy will do long term, then I could fall into the second scenario trap, which is not a desirable scenario and which could have been easily fixed from the very beginning. The first data series can, in probability, reach zero even if its expected outcome is no change, while the second cannot, although, in the long term, it will definitely tend to zero.

**The Minimal Drift**

A case was made that there is a long-term drift in a stock price series that needs to be accounted for and should not be neglected or taken lightly, as in P(t) = μdt + σdw, where μ is the drift. Adding a drift component to the first expression would result in:

Model 3: P(t) = P_{(t-1)} + (RAND() - 0.50 + 0.02)*10

which would generate an upside bias of about 2%. Such an upside bias should be sufficient to make it a game-changer.

**Model 3: Point Upside Bias**

(click to enlarge)

The price series is still unpredictable, and its long-term trend should be easily visible. Extending the price series over a longer time interval would only, in all probability, push prices even higher. It should be noted that having a downside bias would push the price series down over time. For instance, consider the following:

Model 4: P(t) = P_{(t-1)} + (RAND() - 0.50 - 0.02)*10

which would produce a 2% downside bias, something like:

**Model 4: Point Downside Bias**

(click to enlarge)

This would make the point that an upside long-term expectation would make even a random point variation model a suitable long-term trading model. And it would be by looking at the long-term horizon that one could win the game almost by default. This type of strategy would only require having an eye on future prospects of the stock you invest in, and the Buy & Hold would start to look like a reasonable first assumption for a long-term trading strategy.

The last chart shows that there is no need to be long in a stock that has lower future expectations since its price will most certainly follow its diminishing prospects. If, fundamentally, a stock has declining sales and declining profits, it might not look like a long-term positive scenario.

The more you look at it, the more a Buffett-style stock selection is applicable to the problem at hand. Make the best long-term stock selection you can and watch to see if the stock lives up to its expectations. There sure will be trading methods that can help improve on this minimalist design, I know I can design some, therefore why not you.

**The Compounded Drift**

When putting the upside bias as a percentage price move, the picture changes again. This time, everything is accelerated, consider:

Model 5: P(t) = P_{(t-1)} * (1 + (RAND() - 0.50 + 0.02)/10)

where the up bias is again 2%, which would result in a chart like the following:

**Model 5: Percent Upside Bias**

(click to enlarge)

Naturally, a downside bias would destroy a stock more rapidly:

**Model 6: Percent Downside Bias**

(click to enlarge)

Again the long-term prospect should be the main concern. Even a small bias, either to the upside or downside, can make a major difference in the design of a trading strategy that tries to take advantage of a future that is still there to unfold.

Based on these various price modeling functions, it should appear that the most desirable model would be the percent upside bias model (Model 5), which makes long-term prospects increase exponentially in time. In case some might have doubts, they might consider the following long-term chart of the DOW:

**Dow Jones: Long-Term Trend**

(click to enlarge)

which, just by its log scaling, shows that prices have increased at an exponential rate over time. (Sorry, I don't remember where I got that chart).

It's a total game-changer. I find all this just another point made for a Buffett style of playing the game. There is much to learn from his trading/investment methodology. One can build on this basic system and add some trading features, which will tend to push performance levels higher.

Increasing the upside edge to 5% would result in something like this:

**Model 7: 5% Upside Bias **P(t) = P_{(t-1)} * (1 + (RAND() - 0.50 + 0.05)/10)

(click to enlarge)

which really makes it a worthwhile goal.

Then the object of the game becomes to design long-term trading strategies that have this upside edge designed in their very structure. When looking at the above chart, it is clear that the most efficient strategy was to buy from the very beginning and hold for the duration. Nonetheless, I found that it would be even more profitable to trade over a stock accumulation process and use the trading profits to accumulate even more shares. Note that more refined models would require adding fat tails and the use of non-Gaussian price distributions. But these would be refinements and would not change the basic structure presented in this article.

So the conclusion would be that one is bound to find more randomness in designing short-term trading strategies, even for a strategy using Model 5, while designing strategies basically for the long term (using Model 5) is almost an assurance of out-performance just by exercising patience.

Created... April 28, 2013, © Guy R. Fleury. All rights reserved.